#452710

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16060
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,20,9)(4,21,25,41)(5,11,28,38)(6,13)(7,14)(8,12)(10,24,36,29)(16,43)(17,31)(18,33,19,32)(22,30)(23,50)(26,39,27,40)(34,45)(35,54)(37,48)(42,49)(44,58,63,59)(46,56)(47,57)(51,61,64,55)(52,60)(53,62);(1,3,7,19,8,20,6,18)(2,9,14,33,12,15,13,32)(4,22,24,45,25,49,29,16)(5,17,27,47,28,48,26,46)(10,34,21,42,36,43,41,30)(11,37,40,56,38,31,39,57)(23,51,52,63,53,64,54,44)(35,59,50,55,60,58,62,61);(1,4,6,24,8,25,7,29)(2,10,14,41,12,36,13,21)(3,16,18,22,20,45,19,49)(5,23,26,52,28,53,27,54)(9,30,33,43,15,42,32,34)(11,35,40,62,38,60,39,50)(17,44,46,51,48,63,47,64)(31,55,57,59,37,61,56,58);(1,5)(2,11,12,38)(3,17)(4,23)(6,26)(7,27)(8,28)(9,31,15,37)(10,35,36,60)(13,39,14,40)(16,44)(18,46)(19,47)(20,48)(21,50,41,62)(22,51)(24,52)(25,53)(29,54)(30,55,42,61)(32,56,33,57)(34,58,43,59)(45,63)(49,64)

Polycyclic presentation:

2^9|a^2=efgh,b^2=g,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^b=hi,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359906);